The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

Appendix A.
See also Exercise 13.6.
guj75772_ch13.qxd 16/08/2008 03:24 PM Page 472

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Chapter 13
Econometric Modeling: Model Specification and Diagnostic Testing
473
estimated). Now if we add variables to the model, the RSS generally decreases (recall that
as more variables are added to the model, the
R
2
increases), but the degrees of freedom also
decrease because more parameters are estimated. The net outcome depends on whether the
RSS decreases sufficiently to offset the loss of degrees of freedom due to the addition of
regressors. It is quite possible that if a regressor has a strong impact on the regressand—for
example, it may reduce RSS more than the loss in degrees of freedom as a result of its
addition to the model—inclusion of such variables will not only reduce the bias but will
also increase the precision (i.e., reduce the standard errors) of the estimators.
On the other hand, if the relevant variables have only a marginal impact on the regres-
sand, and if they are highly correlated (i.e., VIF is larger), we may reduce the bias in the
coefficients of the variables already included in the model, but increase their standard
errors (i.e., make them less efficient). Indeed, the trade-off in this situation between bias and
precision can be substantial. As you can see from this discussion, the trade-off will depend
on the relative importance of the various regressors.
To conclude this discussion, let us consider the special case where 
r
2 3
=
0, that is, 
X
2
and 
X
3
are uncorrelated. This will result in 
b
3 2
being zero (why?). Therefore, it can be seen
from Eq. (13.3.3) that 
ˆ
α
2
is now unbiased.
11
Also, it seems from Eqs. (13.3.4) and (13.3.5)
that the variances of 
ˆ
α
2
and 
ˆ
β
2
are the same. Is there no harm in dropping the variable 
X
3
from the model even though it may be relevant theoretically? The answer generally is no,
for in this case, as noted earlier, var (
ˆ
α
2
) estimated from Eq. (13.3.4) is still biased and
therefore our hypothesis-testing procedures are likely to remain suspect.
12
Besides, in most
economic research 
X
2
and 
X
3
will be correlated, thus creating the problems discussed
previously.